The third equation in (REF ) as well as the expression of \({\mathcal {U}}^{(n)}(s;z)\)  does not depend on the particular weight we choose and their derivation can be found in [1]}. For the first equation in (REF ), observe that \({\Psi }^{(n)}(s;z)\)  satisfies a Riemann–Hilbert problem with jumps independent on \(z\)  and \(s\) . More precisely, its jumps are localised on the positive real axis (because of the original jumps of \({Y}^{(n)}(s;z)\) ) and, for general \(\alpha >0\) , on the negative real axis, because of the determination of \(z^\alpha \) . Hence, the expression \((\partial _z {\Psi }^{(n)}(s;z)){\Psi }^{(-n)}(s;z)\)  is analytic on \({\mathbb {C}}\mathbb {P}^{1}\setminus \lbrace 0,\infty \rbrace \) , and studying its behaviour at the singular points we can conclude that
\((\partial _z {\Psi }^{(n)}(s;z)){\Psi }^{(-n)}(s;z) = -\frac{1}{2} {\sigma }_3 + \frac{{A}_{-1}^{(n)}(s)}{z} + \frac{{A}_{-2}^{(n)}(s)}{z^2},\) 
